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Unformatted text preview: 1 Discrete Math CS 2800 Prof. Bart Selman [email protected] Module Number Theory 2 The Integers and Division Of course, you already know what the integers are, and what division is… However: There are some specific notations, terminology, and theorems associated with these concepts which you may not know. These form the basics of number theory . – Vital in many important algorithms today (hash functions, cryptography, digital signatures; in general, online security). 3 The divides operator New notation: 3  12 – To specify when an integer evenly divides another integer – Read as “ 3 divides 12 ” The notdivides operator: 5  12 – To specify when an integer does not evenly divide another integer – Read as “5 does not divide 12” 4 Divides, Factor, Multiple Let a , b ∈ Z with a ≠ . Def.: a  b ≡ “ a divides b ” : ≡ ( 5 c ∈ Z : b=ac ) “There is an integer c such that c times a equals b. ” – Example: 3  12 ⇔ True , but 3  7 ⇔ False . Iff a divides b , then we say a is a factor or a divisor of b , and b is a multiple of a . Ex.: “ b is even” :≡ 2 b . Is 0 even? Is −4? 5 Results on the divides operator If a  b and a  c, then a  (b+c) – Example: if 5  25 and 5  30, then 5  (25+30) If a  b, then a  bc for all integers c – Example: if 5  25, then 5  25*c for all ints c If a  b and b  c, then a  c – Example: if 5  25 and 25  100, then 5  100 (“common facts” but good to repeat for background) 6 Divides Relation Theorem: 2200 a , b , c ∈ Z : 1. a 0 2. ( a  b ∧ a  c ) → a  ( b + c ) 3. a  b → a  bc 4. ( a  b ∧ b  c ) → a  c Proof of (2): a  b means there is an s such that b = as , and a  c means that there is a t such that c = at , so b + c = as + at = a ( s + t ), so a ( b + c ) also.■ Corollary: If a, b, c are integers , such that a  b and a  c , then a  mb + nc whenever m and n are integers . 7 More Detailed Version of Proof Show 2200 a , b , c ∈ Z : ( a  b ∧ a  c ) → a  ( b + c ) . Let a , b , c be any integers such that a  b and a  c , and show that a  ( b + c ) . By defn. of  , we know 5 s : b=as , and 5 t : c=at . Let s , t , be such integers. Then b + c = as + at = a ( s + t ) , so 5 u : b + c = au , namely u = s + t . Thus a ( b + c ) . Divides Relation Corollary: If a, b, c are integers , such that a  b and a  c , then a  mb + nc whenever m and n are integers . Proof: From previous theorem part 3 (i.e., ab → abe) it follows that a  mb and a  nc ; again, from previous theorem part 2 (i.e., ( a  b ∧ a  c ) → a  ( b + c )) it follows that a  mb + nc 9 The Division “Algorithm” Theorem: Division Algorithm  Let a be an integer and d a positive integer. Then there are unique integers q and r, with 0 ≤r < d, such that a = dq+r....
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 SELMAN

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